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Conference Programme

Monday, 13. 9. 2021

1300 Lunch

1400

Arrival and registration
1630 Invitation coffee break
  Chairmans, Daniela Velichová, Miroslav Lávička

1700

Daniela Velichová: 30 Years of Seminars on Geometry and Graphics in Slovakia

1720

Invited lecture: prof. RND. Ján Čižmár, PhD.
Actual Message of the Euclidean Elements

  Poster section
  Margita Vajsáblová: Geometric Aspects in Mathematical Foundations of Cartography

1800

Dinner

1900

Get-together party

Tuesday, 14. 9. 2021

800 Breakfast
  Chairman: Pavel Pech
900 Invited lecture: RNDr. Michal Bizzarri, PhD.
Geometry and Tool Motion Planning for Curvature Adapted CNC Machining
1000 Coffee break
  Chairwoman: Daniela Richtáriková
1030 Hellmuth Stachel: Billiard Motions in Ellipses -- Invariants of Projective Nature
1050 Pavel Pech: Chord of Conics
1110 Miroslav Lávička: Note on Curves of Tschirnhaus Type
1130 Jan Vršek: Symmetries of Discrete Curves Via Trigonometric Interpolation
1200 Lunch

Chairman: Pavel Chalmovianský

1400

Invited lecture: Mgr. Michal Piovarči, PhD.
Perception-Aware Computational Fabrication

1500 Alexej Kolcun: Fergusonova krivka v príkladoch
1520 Marta Hlavová: Interpolation of Transcendental Curve Points
1540 Marcel Makovník: Mesh Refinement Near Singularities
1600 Coffee break
  Chairman: Jan Safařík
1630 Dana Kolářová: Plocha šikmého průchodu v nosné konstrukci historických mostů
1650 Tatiana Rückschlossová, Margita Vajsáblová: Geometria v profile absolventa stavebného odboru do roku 2030
1710 Michaela Holešová: Constructions of Quatrefoil

1800

Dinner

1900

Meetings of societies SSGG, ČSGG

Wednesday, 15. 9. 2021

800 Breakfast
  Chairman: Jan Vršek
900 Pavel Chalmovianský: What is a Curve?
920 Alžbeta Mackovová: Voronoi Diagram in Three-Dimensional Hyperbolic Space
940 Adriana Bosáková: A Note on the Structure of the Intersection Multiplicity of Two Plane curves

1000

Coffee break
  Chairwoman: Margita Vajsáblová

1030

Invited lecture: prof. Gunter Weiss
Geometric Problems that Take Possesion of Me

1130 Michal Zamboj: Twisted Filaments with Polyhedral Symmetries
1200 Lunch
1300 Excursion

1900

Conference dinner

Thursday, 16. 9. 2021

800 Breakfast
  GeoGebra workshop
  Chairmans: Michaela Holešová, Roman Hašek
 900 Věra Ferdiánová: GeoGebra nástroje ve výuce planimetrie
 920 Marie Chodorová: Kolineace kuželoseček v GeoGebře
 940 Jakub Řada: GeoGebra Tools for Drawing in Double Orthogonal Projection and 4D Perspective
 1000 Roman Hašek: Dynamic Geometry Online
 1020 Jan Šafařík, Jana Bulantová, Lucie Zrůstová: Sbírka řešených příkladů z konstruktivní geometrie
1130 Lunch
1300 Departure

 

Abstracts


Geometry and Tool Motion Planning for Curvature Adapted CNC Machining

BIZZARRI Michal

Katedra matematiky Fakulta aplikovaných věd, Západočeská univerzita v Plzni
Plzeň, Česká republika
e-mail: bizzarri@ntis.zcu.cz

CNC machining is the leading subtractive manufacturing technology. Although it is in use since decades, it is far from fully solved and still a rich source for challenging problems in geometric computing. We demonstrate this at hand of 5-axis machining of freeform surfaces, where the degrees of freedom in selecting and moving the cutting tool allow one to adapt the tool motion optimally to the surface to be produced. We aim at a high-quality surface finish, thereby reducing the need for hard-to-control post-machining processes such as grinding and polishing. Our work is based on a careful geometric analysis of curvature-adapted machining via so-called second order line contact between tool and target surface. On the geometric side, this leads to a new continuous transition between “dual” classical results in surface theory concerning osculating circles of surface curves and osculating cones of tangentially circumscribed developable surfaces. Practically, it serves as an effective basis for tool motion planning. Unlike previous approaches to curvature-adapted machining, we solve locally optimal tool positioning and motion planning within a single optimization framework and achieve curvature adaptation even for convex surfaces. This is possible with a toroidal cutter that contains a negatively curved cutting area. The effectiveness of our approach is verified at hand of digital models, simulations and machined parts, including a comparison to results generated with commercial software.


A Note on the Structure of the Intersection Multiplicity of Two Plane curves

BOSÁKOVÁ Adriana

Fakulta matematiky, fyziky a informatiky, Univerzita Komenského v Bratislave
Bratislava, Slovakia
e-mail: adriana.bosakova@fmph.uniba.sk

Intersection multiplicity is an invariant in algebraic geometry, which describes the complexity of an intersection of algebraic varieties. Studying its internal structure can help us describe the algebraic and geometric properties of such intersection. We focus on the case of the intersection of two plane affine algebraic curves. By existing results, we know that the intersection multiplicity is greater or equal to (mn + t), where m and n are the multiplicities of the intersection point on the individual curves and t is the number of their common tangents at this point. In this paper we show an improved version of this result.


Sbírka řešených příkladů z konstruktivní geometrie

BULANTOVÁ Jana, ŠAFAŘÍK Jan, ZRŮSTOVÁ Lucie

FAST VUT Brno
Brno, ČR
e-mail: bulantova.j@fce.vutbr.cz

Cílem příspěvku je seznámit s novou online sbírkou řešených příkladů z konstruktivní geometrie, která vznikla během distanční výuky v AR 2020/2021 na Stavební fakultě VUT v Brně především za pomocí GeoGebry, počítačového programu pro interaktivní geometrii.


What is a Curve?

CHALMOVIANSKÝ Pavel

Department of Algebra and Geometry, Faculty of Mathematics and Physics, Comenius University
Bratislava, Slovakia
e-mail: Pavel.Chalmoviansky@fmph.uniba.sk

We look at the notion of curve from different points of view.
Historically, the notion was developed using functions of one variable and using the notion of dimension. Both requirements are necessary and, moreover, the numbers we allow to work with play an important role what can be a curve.


Kolineace kuželoseček v GeoGebře.

CHODOROVÁ Marie

Katedra algebry a geometrie, Přírodovědecká fakulta Univerzity Palackého v Olomouci
Olomouc, Česká Republika
e-mail: marie.chodorova@upol.cz

Cílem příspěvku je ukázat na příkladech konstrukce kuželoseček pomocí kolineace, kdy kolineákním obrazem kuželosečky je kružnice (příp. jiná kuželosečka). Tyto konstrukce mají praktické využití při sestrojování řezů rotačních a zejména nerotačních kuželů. Konstrukce jsou zpracovány v GeoGebře formou pracovních listů.


Aktuálny odkaz Euklidových Základov

ČIŽMÁR Ján
geometer emeritus
Bratislava, Slovensko
e-mail: jan.cizmar85@gmail.com

Príspevok uvádza krátky prehľad obsahu trinástich kníh kardinálneho Euklidovho diela Základy (Stoicheia) z hľadiska dnešnej klasifikácie matematických odborov. Dielo je po prvý raz v úplnosti pripravené do tlače v slovenskom preklade autora príspevku a je opatrené stručným komentárom približujúcim obsah diela v jazyku dnešnej matematiky.

Dôležitým zámerom príspevku je poukázať na niekoľkovrstvový význam Euklidovej odbornej terminológie, ktorý z pohľadu dnešných zásad logiky a presnosti odborného jazyka vyžaduje hlbšie zásahy do tradičnej rutiny v používaní matematickej terminológie v písomných prejavoch, najmä v tvorbe učebníc, i v dennej didaktickej praxi.

Druhým významným trvalo aktuálnym odkazom Euklida, skryto prestupu-júcim celým jeho dielom, no osobitne Základmi, je zdôrazňovanie sústavnej primeranej didaktickej axiomaticko-deduktívnej metódy vo vyučovaní matematiky (a nielen matematiky).


GeoGebra Nástroje ve výuce planimetrie

FERDIÁNOVÁ Věra

KMA, PřF, Ostravská univerzita
Ostrava, Czech republic
e-mail: vera.ferdianova@gmail.com

Cílem příspěvku bude představení tvorby GeoGebra nástrojů, jenž napomáhají při běžných planimetrických konstrukcí.


Dynamická geometrie online / Dynamic Geometry Online

HAŠEK Roman

Jihočeská univerzita v Českých Budějovicích, Pedagogická fakulta
České Budějovice, Česká republika
e-mail: hasek@pf.jcu.cz

Příspěvek je zaměřen na online prezentaci matematického a geometrického obsahu. Věnuje se konkrétnímu způsobu prezentace tohoto obsahu prostřednictvím materiálů ve formátu HTML, které mohou mít jak podobu samostatné webové stránky, tak i dílčí součásti nějakého online vzdělávacího systému, například Moodle. Na vybraných příkladech bude ukázáno, jak lze použitím knihoven JavaScriptu JSXGraph a MathJax vytvořit kvalitní prezentaci geometrického obsahu, v níž se volně kombinuje matematický text ve formátu LaTeX s interaktivními dynamickými obrázky.

The contribution deals with the online presentation of mathematical and geometric content. Specifically, it focuses on a way how to present this content through HTML materials, which can take the form of both stand-alone websites and sub-components of an online education system, such as Moodle. Selected examples will show the use of JavaScript libraries JSXGraph and MathJax to create a quality presentation of geometric content. The benefit of using these libraries is the opportunity to freely combine mathematical text in LaTeX format with interactive dynamic images in these materials.


Interpolation of Transcendental Curve Points/ Interpolace bodů transcendentní křivky

HLAVOVÁ Marta

ČVUT v Praze, Fakulta strojní
Praha, ČR
e-mail: marta.hlavova@fs.cvut.cz

This contribution deals with the dependence of geometrical accuracy of a natural cubic spline on the Euclidean distance between the definition points. As an example, a transcendental curve - a graph of function f: y = sin(x), x from the interval [0,2Pi] - is used.

Příspěvek je věnován vztahu mezi geometrickou přesností interpolace bodů transcendentní křivky přirozeným kubickým splinem a přímou vzdáleností definičních bodů. Jako modelová transcendentní křivka je zde použita křivka grafu funkce f: y = sin(x) na intervalu [0,2Pi].


Constructions of Quatrefoil

HOLEŠOVÁ Michaela

KSMAM ŽU v Žiline
Žilina, Slovenská republika
e-mail: michaela.holesova@uniza.sk

In architecture, we meet with different and also more complex shapes. Due to the technical construction, they are usually made up of basic geometric elements, such as a line and a circle. They are used to create interesting floor plans, openings and decorative elements. In addition to the relatively frequently used ovals, we also encounter the so-called quatrefoil based on four circles or ovals in Slovak architecture. These are arranged symmetrically and are connected to each other by lines which are predominantly tangential to these circles. We will show some interesting constructions of quatrefoils.


Plocha šikmého průchodu v nosné konstrukci historických mostů

KOLÁŘOVÁ Dana

Fakulta architektury ČVUT v Praze
Praha 6, Česká republika
e-mail: kolarova@fa.cvut.cz

Příspěvek představuje vývoj použití plochy šikmého průchodu v nosné konstrukci historických mostů.


Fergusonova krivka v príkladoch / Ferguson Curve in the Examples

KOLCUN Alexej

Katedra informatiky a počítačů Přírodovědecká fakulta Ostravská univerzita
Ostrava, Česko
e-mail: alexej.kolcun@osu.cz

Pri demonštrácií vzájomného vzťahu riadiacich prvkov a výsledného tvaru parametrických kriviek sa učebnice spravidla obmedzujú na kvalitatívnu charakterizáciu tohto vzťahu. V príspevku formulujeme niekoľko jednoduchých úloh, kde pre Fergusonovu krivku nachádzame detailnejšie – kvantitatívne vzťahy medzi riadiacimi elementami a výsledným tvarom krivky.

When demonstrating the relationship between control elements and the resulting shape of parametric curves, textbooks are usually limited to the qualitative characterization of this relationship. In this paper, we formulate a few simple tasks, where we find more detailed - quantitative relationships between the control elements and the resulting shape for the Ferguson curve.


Note on Curves of Tschirnhaus Type

LÁVIČKA Miroslav

Katedra matematiky Fakulta aplikovaných věd Západočeská univerzita v Plzni
Plzeň, Czech Republic
e-mail: lavicka@kma.zcu.cz

Special Pythagorean hodograph curves which can be considered, from construction point of view, as degree n generalizations of the famous Tschirnhaus cubic will be presented. Their properties will be discussed. In addition, it will be proved that for each n there exists only one curve of Tschirnhaus type up to similarities.


Voronoi Diagram in Three-Dimensional Hyperbolic Space

MACKOVOVÁ Alžbeta

Katedra algebry a geometrie, Fakulta matematiky, fyziky a informatiky, Univerzita Komenského v Bratislave
Bratislava, Slovensko
e-mail: alzbeta.mackovova@gmail.com

Voronoi diagrams belong to favorite structures in computational geometry. In our work, we focus on construction and properties of Voronoi diagram in three-dimensional hyperbolic space represented by Poincaré ball model. We first define and illustrate the basic geometric elements of Poicaré ball model that we will work with. Finally, we construct a Voronoi diagram and describe some of its properties, which depend on the position of the generators.


Mesh Refinement Near Singularities

MAKOVNÍK Marcel

Katedra algebry a geometrie Fakulta matematiky, fyziky a informatiky Univerzita Komenského v Bratislave
Bratislava, SR
e-mail: marcel.makovnik@fmph.uniba.sk

A singular point of an implicit surface is a point, whose gradient is a zero vector. Mesh refinement via quadric fitting takes point-normal pairs as an input. Hence, singularities in the mesh may be obtained by assigning zero normal vector to the input point. In this paper we inspect on the behaviour of the refinement near the singularity with respect to connectivity of the mesh. We provide examples of refinement on synthetic and real data.


Chord of Conics

PECH Pavel

Jihočeská univerzita v Č. Budějovicích, Pedagogická fakulta
České Budějovice, ČR
e-mail: pech@jcu.cz

The paper deals with locus of points related to chords of conic sections. Firstly the locus is explored using dynamic geometry software GeoGebra, secondly using
computer algebra system CoCoA by elimination the locus equation is derived.
The locus is related to well-known theorems and concepts, as Simson line and Frégier point.


Perception-Aware Computational Fabrication

PIOVARČI Michal

Institute of Science and Technology Austria
Klosterneuburg, Austria
e-mail: michael.piovarci@ist.ac.at

Haptic and visual feedback are important for assessing objects´ quality and affordance. One of the benefits of additive manufacturing is that it enables the creation of objects with personalized tactile and visual properties. This personalization is realized by the ability to deposit functionally graded materials at microscopic resolution. However, faithfully reproducing real-world objects on a 3D printer is a challenging endeavor. A large number of available materials and freedom in material deposition make exploring the space of printable objects difficult. Furthermore, current 3D printers can perfectly capture only a small amount of objects from the real world which makes high-quality reproductions challenging. Interestingly, similar to the manufacturing hardware, our senses of touch and sight have inborn limitations given by biological constraints. In this talk, I will demonstrate how we can leverage the limitations of the Human Sensorial System to increase the apparent gamut of a 3D printer by combining numerical optimization with perceptual insights. Instead of optimizing for exact replicas, we search for perceptually equivalent solutions. I will show applications of the proposed methodology to designing objects with prescribed compliance, mimicking the haptics of drawing tools, and manufacturing objects with spatially varying gloss.


GeoGebra Tools for Drawing in Double Orthogonal Projection and 4D Perspective

ŘADA Jakub

Matematický ústav UK, Matematicko-fyzikální fakulta, Univerzita Karlova, Praha
Praha, Česká republika
e-mail: radajaku@seznam.cz

With the advent of modern technology, it is often much more accessible to draw on a computer than on paper. It is necessary to choose the right program. There are efforts to make special software for specific projection, but usually, these programs do not last long. For example, Petr Plavjanik´s program - Deskriptivní geometrie. In 1999, the author celebrated success with it in SPA and INVEX ´99. After a few years, its development was stopped. For this reason, it is better and more convenient for the users to use some general graphic software. GeoGebra is one of them, and it also has an outstanding possibility to create your own custom tools. Each projection has its principles for which custom tools can be created. This article aims to show using GeoGebra with own custom tools as additional software to display objects in double orthogonal projection and 4D perspective.


Geometria v profile absolventa stavebného odboru do roku 2030

RÜCKSCHLOSSOVÁ Tatiana

Katedra matematiky a deskriptívnej geometrie, Stavebná fakulta STU v Bratislave
Bratislava, SR
e-mail: tatiana@math.sk

Študijné programy na Stavebnej fakulte, Fakulte architektúry a dizajnu a na Ústave manažmentu STU v Bratislave sú po akreditácii 2021 naďalej otvorené geometrickým predmetom. Stavebný inžinier, architekt, či geodet budúcnosti má veľké softvérové a technologické možnosti v tvorbe a realizácii stavebného objektu, pričom geometrická a priestorová predstavivosť umocňuje ich efektívne využitie. Obsah, metódy a formy výučby geometrie si vyžadujú aktualizáciu súvisiacu s následnou aplikáciou vedomostí študenta v odborných predmetoch a v stavebnej praxi. V príspevku ukážeme tieto aspekty na aplikáciách geometrie vo výučbe geometrických predmetov pre konkrétne študijné odbory. Zároveň uvedieme analýzu, skúsenosti a ukážky metód dištančnej formy výučby geometrických predmetov.


Billiard Motions in Ellipses -- Invariants of Projective Nature

STACHEL Hellmuth

Vienna University of Technology Institute of Discrete Mathematics and Geometry
Wien, Austria
e-mail: stachel@dmg.tuwien.ac.at

A billiard is the trajectory of a mass point in a domain with ideal physical reflections in the boundary. Already for two centuries, billiards in ellipses have attracted the attention of mathematicians, beginning with J.-V. Poncelet and C.G.J. Jacobi. Recent computer animations, which were carried out by D. Reznik (Brazil), stimulated a new vivid interest on this well studied objects.

The sides of any billiard in an ellipse e are tangent to a confocal ellipse or hyperbola c, called caustic. If one billiard in e with caustic c closes after N reflections, then it closes for each choice of the initial vertex. When this vertex varies on e, then this defines a socalled billiard motion, though this variation neither preserves angles or side lengths nor is a projective motion. However, the total length of the periodic billiard is invariant, and Reznik identified about 80 invariants.

The goal of this presentation is to recall some metric invariants and to present a few projective invariants which partly date back to Poncelet, for example the envelopes of the diagonals.


Geometric Aspects in Mathematical Foundations of Cartography

VAJSÁBLOVÁ Margita

Katedra matematiky a deskriptívnej geometrie, Stavebná fakulta STU v Bratislave
Bratislava, Slovenská republika
e-mail: margita.vajsablova@stuba.sk

Cartographic projection represents the relationship between the reference surface of the Earth and its map image. In essence, it has a geometric and mathematical character. In this paper, the emphasis is on the presentation of geometric aspects in the selection and creation of cartographic projection, namely the shape of the reference surface of the Earth, geometric characteristics of the projected area, curves on reference surfaces and image shape requirements of projected elements. These aspects are included, although often hidden in the university textbook Mathematical Foundations of Cartography, in the creation of which I applied my mathematical logic, geometric eyes and cartographic heart.


30 Years of Seminars on Geometry and Graphics in Slovakia

VELICHOVÁ Daniela

ÚMF SjF STU
Bratislava, SR
e-mail: daniela.velichova@stuba.sk

A brief look back and an overview of the 30-years history of professional seminars on geometry and graphics held in Slovakia from the year 1991 is presented in this article. Seminars have developed into the international Symposiums on Computer Geometry and they were organized by the Slovak Society for Geometry and Graphics until 2014. These international conferences were joined from the year 2015 with the Conference on Geometry and Graphics held annually in the Czech Republic by the Czech Society for Geometry and Graphics. Successful 6 meetings of Slovak and Czech scientific communities active in geometry and computer graphics organised alternately in the Czech Republic and Slovakia proved that this idea was useful and fruitful. The 7th common Slovak-Czech Conference on Geometry and Graphics took place in Kočovce, Slovakia, in September 2021.


Symmetries of Discrete Curves Via Trigonometric Interpolation

VRŠEK Jan

Západočeská univerzita v Plzni
Plzeň, Česká republika
e-mail: vrsekjan@kma.zcu.cz

The talk will be devoted to the computation of symmetries of a~closed discrete curve in Euclidean plane. We will show how to replace discrete curve by the so called trigonometric curve, i.e., curve possessing a parametrization in terms of sines and cosines. The curve is chosen so that its symmetry group contains the symmetry group of the original discrete curve. We demonstrate that the symmetries of a trigonometric curve can be found easily, which we use to solve the original problem. We conclude the talk by observing how convex hulls, together with the previous approach, allow us to find symmetries of clouds of points.


Geometric Problems that Take Possession of Me

WEISS Gunter

retired
Vienna, Austria
e-mail: weissgunter@gmx.at

The lecture deals with my way of doing research: It often starts with getting stimulated by an article or a lecture at a conference. Putting one of its details in another context, mostly a projective geometric one, and then, by looking for meaningful generalisations, it is finally possible to extract a general principle which brings together divergent and often well-known facts. The resulting aha moments are almost addictive. Examples of such topics will be “polyhedrons made by setsquares”, “C^r-biarcs of circles and conics”, “Stellae octangulae”, “Frégier’s theorem”, “Minkowski Geometry resp. metric planes”. There remain still many open questions, which could be perhaps answered by researchers having another scientific background than mine. Unfortunately the mentioned topics are not within the actual scientific mainstream, but their inner beauty justifies to be curious and to struggle for getting answers to open questions.


Twisted Filaments with Polyhedral Symmetries

ZAMBOJ Michal

Department of Mathematics and Mathematical Education, Faculty of Education, Charles University
Prague, Czech Republic
e-mail: michal.zamboj@pedf.cuni.cz

Twisted filaments are common structures in nature. We describe a geometric method of their creation, such that they possess a predefined polyhedral symmetry. We start from a well-chosen polyhedron mapped to a circumscribed 2-sphere. The images of the vertices on the 2-sphere create circular fibers in a 3-sphere in the Hopf fibration. Moreover, circles around the vertices form torus filaments around the fibers. After all, we visualize the filaments inside of the 3-sphere in a double-orthogonal and stereographic projection.